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Two closely related problems are posed with respect to the structure of strictly stationary processes. Such a representation is comparatively rare for a finite state Markov chain. This is shown to be the case for every ergodic finite state Markov chain with nonperiodic states Section 4.
Sufficient conditions for such representations in the case of a general strictly stationary process are obtained in Section 5. The Fourier coefficients. Notice that the block Toeplitz matrix An is generally not Toeplitz. Limit properties of the convolution sequence of a regular measure on a compact topological semigroup are examined in this paper. Recently Bellman  and GRenaNdeR  considered special limit theorems for products of independent identically distributed random operators.
Such problems are closely related to those in this paper. It should be noted that similar questions arise when considering the structure of stationary stochastic processes . Various results on compact semigroups are used in characterizing the class of limit measures , , . A stochastic process is commonly used as a model in studying the behavior of a random system through time.
Processes of independent random variables are the simplest and most completely understood, It is, however, clear that these axe extremely limited in scope as models and one must have recourse to dependent processes the random variables xt not independent in order to have any power in description.
For simplicity, let us further restrict ourselves to processes that are stable through time, stationary processes. For such processes the probabilities of events shifted through time remain the same, that is, the probability. The results obtained in this chapter may be of some interest from the point of view of analysis. However, they have an immediate interpretation in terms of certain representation theorems for stationary random processes on a finite time interval, and this provided part of the motivation for the investigation.
Selected Works of Murray Rosenblatt | itocagawoler.ga
Our interest is in finite interval translation kernel integral equation eigenvalue problems, that is, in the integral equation. By this we mean that every complex-valued element F. Recently interest has arisen in statistical applications of the bispectrum of stationary random processes.
The bispectrum can be thought of as the Fourier transform of the third-order moment function of the process. The principal area of statistical harmonic analysis to receive attention previous to this time has been second-order i. A desire to study phenomena of a nonlinear character has attracted attention to the higher order theory. Such was the case, for example, in a recent study by Hasselmann, Munk and MacDonald  where the bispectrum is used in connection with oceanographic problems, among which, as the authors state, a number of interesting phenomena such as surf beats, wave breaking, and the energy transfer between wave components can be explained only by the nonlinearity of the wave motion.
The bispectrum therefore provides a first glimpse at the nonlinear effects. It is the purpose of the present paper to discuss estimating the weighted and unweighted bispectral density given a set of observations of the process. The relevant properties consistency and asymptotic unbiasedness of the estimates are derived for certain general classes of processes.
A harmonic analysis of the solutions is carried out and the form of the spectrum is estimated for large time. A similar analysts is also carried out for an irrotaticnal solulion of a similar 3-dimensionat system of equations. For large time and moderate wavenumbers there is. Comments are also made on the asymptotic distribution of a class of random solutions. Estimates of the density function of a population based on a sample of independent observations have been considered in a number of papers [1,].
Questions of bias, variance and asymptotic distribution of the estimates have been dealt with at greatest length. Our object is to look at such estimates of the density function when the observations are dependent. The results will not be dealt with in the most general context or under very general conditions.
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To obtain results in a simple and readily understandable form, the observations are assumed to be sampled from a stationary Markov sequence with a fairly strong condition on the Markov transition operator. However, the extent to which some of the conditions can be obviously relaxed will be indicated. There is a large class of problems in which the estimation of curves arises naturally see , . It is curious that one of the earliest extensive investigations of this type involves the estimation of the spectral density function when sampling from a stationary sequence , , , .
Even though the simple histogram has been used for years, it was only later that the simpler question of estimating a probability density function was dealt with at some length , , .
Because the final character of the usual results obtained in both problem areas is quite similar, and the arguments are much more transparent in the case of the probability density function, we shall develop the results for the probability density function first. Later some corresponding results for spectra will be given. We consider density estimates of the usual type generated by a weight function.
Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity.
Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved. Lii and M. Asymptotic behavior of a spline estimate of a density function. Reprinted with permission of Elsevier Inc.
A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Some related remarks are made about random solutions of the Burgers equation. A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian.
In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.
Limit theorems with a non-Gaussian in fact nonstable limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence. Analogous limit theorems are here obtained for finite Fourier transforms of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence. NonGaussian linear processes are considered. It is shown that the phase of the transfer function can be estimated under broad conditions.
This is not true of Gaussian linear processes and in this sense Gaussian linear processes are atypical. The asymptotic behavior of a phase estimate is determined. The phase estimates make use of bispectral estimates. These ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum phase.
A number of computational illustrations are given. Asymptotic normality is proven for spectral density estimates assuming strong mixing and a limited number of moment conditions for the process analyzed. The result holds for a large class of processes that are not linear and does not require the existence of all moments. We consider the problem of estimating the filter generating a non-Gaussian linear process and the decon-voiution of that process when the spectral density of the process has zeros. Without using a minimum phase assumption we show that often if there are only finitely many zeros there are procedures to effect such an estimation and deconvolution.
Solutions of the Burgers equation with a stationary spatially stochastic initial condition are considered. A class of limit laws for the solution which correspond to a scale renormalization is considered. This paper discusses a least-squares procedure and the use of the periodogram for isolating a discrete harmonic of a time series. It is shown that the usual asymptotics on estimation of frequency, amplitude and phase of such a harmonic have to be used with great caution from a moderate sample perspective.
Computational issues are discussed and some illustrations are provided. Breidt, R. Davis, K. Lii, and M. Maximum likelihood estimation for noncausal autoregressive processes. Multivariate Anal.
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